Optimal. Leaf size=94 \[ -\frac{2 b^2 (c+d x)^{3/2} (b c-a d)}{d^4}+\frac{6 b \sqrt{c+d x} (b c-a d)^2}{d^4}+\frac{2 (b c-a d)^3}{d^4 \sqrt{c+d x}}+\frac{2 b^3 (c+d x)^{5/2}}{5 d^4} \]
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Rubi [A] time = 0.0297938, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ -\frac{2 b^2 (c+d x)^{3/2} (b c-a d)}{d^4}+\frac{6 b \sqrt{c+d x} (b c-a d)^2}{d^4}+\frac{2 (b c-a d)^3}{d^4 \sqrt{c+d x}}+\frac{2 b^3 (c+d x)^{5/2}}{5 d^4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^3}{(c+d x)^{3/2}} \, dx &=\int \left (\frac{(-b c+a d)^3}{d^3 (c+d x)^{3/2}}+\frac{3 b (b c-a d)^2}{d^3 \sqrt{c+d x}}-\frac{3 b^2 (b c-a d) \sqrt{c+d x}}{d^3}+\frac{b^3 (c+d x)^{3/2}}{d^3}\right ) \, dx\\ &=\frac{2 (b c-a d)^3}{d^4 \sqrt{c+d x}}+\frac{6 b (b c-a d)^2 \sqrt{c+d x}}{d^4}-\frac{2 b^2 (b c-a d) (c+d x)^{3/2}}{d^4}+\frac{2 b^3 (c+d x)^{5/2}}{5 d^4}\\ \end{align*}
Mathematica [A] time = 0.0544484, size = 78, normalized size = 0.83 \[ \frac{2 \left (-5 b^2 (c+d x)^2 (b c-a d)+15 b (c+d x) (b c-a d)^2+5 (b c-a d)^3+b^3 (c+d x)^3\right )}{5 d^4 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 116, normalized size = 1.2 \begin{align*} -{\frac{-2\,{b}^{3}{x}^{3}{d}^{3}-10\,a{b}^{2}{d}^{3}{x}^{2}+4\,{b}^{3}c{d}^{2}{x}^{2}-30\,{a}^{2}b{d}^{3}x+40\,a{b}^{2}c{d}^{2}x-16\,{b}^{3}{c}^{2}dx+10\,{a}^{3}{d}^{3}-60\,{a}^{2}bc{d}^{2}+80\,a{b}^{2}{c}^{2}d-32\,{b}^{3}{c}^{3}}{5\,{d}^{4}}{\frac{1}{\sqrt{dx+c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965714, size = 169, normalized size = 1.8 \begin{align*} \frac{2 \,{\left (\frac{{\left (d x + c\right )}^{\frac{5}{2}} b^{3} - 5 \,{\left (b^{3} c - a b^{2} d\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 15 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt{d x + c}}{d^{3}} + \frac{5 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}}{\sqrt{d x + c} d^{3}}\right )}}{5 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1117, size = 259, normalized size = 2.76 \begin{align*} \frac{2 \,{\left (b^{3} d^{3} x^{3} + 16 \, b^{3} c^{3} - 40 \, a b^{2} c^{2} d + 30 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3} -{\left (2 \, b^{3} c d^{2} - 5 \, a b^{2} d^{3}\right )} x^{2} +{\left (8 \, b^{3} c^{2} d - 20 \, a b^{2} c d^{2} + 15 \, a^{2} b d^{3}\right )} x\right )} \sqrt{d x + c}}{5 \,{\left (d^{5} x + c d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.0578, size = 109, normalized size = 1.16 \begin{align*} \frac{2 b^{3} \left (c + d x\right )^{\frac{5}{2}}}{5 d^{4}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (6 a b^{2} d - 6 b^{3} c\right )}{3 d^{4}} + \frac{\sqrt{c + d x} \left (6 a^{2} b d^{2} - 12 a b^{2} c d + 6 b^{3} c^{2}\right )}{d^{4}} - \frac{2 \left (a d - b c\right )^{3}}{d^{4} \sqrt{c + d x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06752, size = 205, normalized size = 2.18 \begin{align*} \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}}{\sqrt{d x + c} d^{4}} + \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{5}{2}} b^{3} d^{16} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} b^{3} c d^{16} + 15 \, \sqrt{d x + c} b^{3} c^{2} d^{16} + 5 \,{\left (d x + c\right )}^{\frac{3}{2}} a b^{2} d^{17} - 30 \, \sqrt{d x + c} a b^{2} c d^{17} + 15 \, \sqrt{d x + c} a^{2} b d^{18}\right )}}{5 \, d^{20}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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